Sure, let's solve the equation [tex]\( x = \frac{2}{3} \pi r^3 \)[/tex] for [tex]\( r \)[/tex] step-by-step:
1. Start with the given equation:
[tex]\[
x = \frac{2}{3} \pi r^3
\][/tex]
2. Isolate [tex]\( r^3 \)[/tex]:
To isolate [tex]\( r^3 \)[/tex], multiply both sides of the equation by the reciprocal of the coefficient of [tex]\( r^3 \)[/tex], which is [tex]\(\frac{3}{2 \pi}\)[/tex]:
[tex]\[
x = \frac{2}{3} \pi r^3 \implies x \cdot \frac{3}{2 \pi} = \left( \frac{2}{3} \pi r^3 \right) \cdot \frac{3}{2 \pi}
\][/tex]
The right side simplifies to [tex]\( r^3 \)[/tex]:
[tex]\[
\frac{3x}{2\pi} = r^3
\][/tex]
3. Solve for [tex]\( r \)[/tex]:
To solve for [tex]\( r \)[/tex], take the cube root of both sides of the equation:
[tex]\[
r = \sqrt[3]{\frac{3x}{2\pi}}
\][/tex]
4. Determine the correct option:
We need to compare this derived expression with the given options:
- Option A: [tex]\( r = \sqrt[3]{\frac{3 x}{2 \pi}} \)[/tex]
- Option B: [tex]\( r = \sqrt[3]{3 x - 2 \pi} \)[/tex]
- Option C: [tex]\( T = \sqrt[3]{\frac{2 \pi}{3 x}} \)[/tex]
- Option D: [tex]\( T = \sqrt[3]{3 x (2 \pi)} \)[/tex]
Clearly, Option A matches our derived expression exactly.
Therefore, the correct solution is:
A. [tex]\( r = \sqrt[3]{\frac{3 x}{2 \pi}} \)[/tex]
